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<b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:51"/><i><font color="Green">E19</font></i>: 
( 1 <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 ;<br/>

<b>let </b><font color="Maroon">fi</font> be   <a href="ordinal2.html#NM1">Ordinal-Sequence</a>;<br/>

<b>assume </b><a NAME="E2:51"/><i><font color="Green">E21</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> ) )
 ;<br/>

<a NAME="E3:51"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">fi</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#T54">ORDINAL2:54</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:51_1"/><i><font color="Green">E35</font></i>: 
 <a href="ordinal2.html#K4">sup</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:51_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">fi</font> &amp;  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">fi</font></span>)</span> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E12:51_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal2.html#T35">ORDINAL2:35</a></i>;<br/><b>then consider </b><font color="Maroon">D</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:51_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fi</font> &amp; <font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/><a NAME="E5:51_1"/>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">fi</font>
 <b>by </b><i>, , <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/><a NAME="E6:51_1"/><b>then </b>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E12:51_1"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a>, <a class="ref" href="ordinal2.html#T1">ORDINAL2:1</a></i>;<br/><a NAME="E7:51_1"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E8:51_1"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T12">ORDINAL1:12</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:51_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E12:51_1"/><i><font color="Green">E39</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E2:51_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ordinal3.html#E4:51_1"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/><a NAME="E13:51_1"/><b>then </b>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> &amp; <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">x</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> &amp; <font color="Maroon">C</font> <a href="ordinal2.html#K10">+^</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font> &amp; <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">x</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E2:51_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ordinal3.html#E4:51_1"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a>, <a class="ref" href="ordinal2.html#T53">ORDINAL2:53</a></i>;<br/><a NAME="E14:51_1"/><b>then </b>
( <font color="Maroon">C</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fi</font> &amp;  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">fi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E12:51_1"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="ordinal2.html#T48">ORDINAL2:48</a></i>;<br/><b>hence </b><a NAME="E15:51_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:51"/><b>then </b>
 <a href="ordinal2.html#K4">sup</a> <font color="Maroon">fi</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>
<b>hence </b><a NAME="E6:51"/>
<font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">fi</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#D6">ORDINAL1:def 6</a></i>;<br/>


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